Misplaced Pages

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73-422: The interior of S is the complement of the closure of the complement of S . In this sense interior and closure are dual notions. The exterior of a set S is the complement of the closure of S ; it consists of the points that are in neither the set nor its boundary . The interior, boundary, and exterior of a subset together partition the whole space into three blocks (or fewer when one or more of these

146-415: A natural topology since it is locally Euclidean. Similarly, every simplex and every simplicial complex inherits a natural topology from . The Sierpiński space is the simplest non-discrete topological space. It has important relations to the theory of computation and semantics. Every subset of a topological space can be given the subspace topology in which the open sets are the intersections of

219-489: A set X may be defined as a collection τ {\displaystyle \tau } of subsets of X , called open sets and satisfying the following axioms: As this definition of a topology is the most commonly used, the set τ {\displaystyle \tau } of the open sets is commonly called a topology on X . {\displaystyle X.} A subset C ⊆ X {\displaystyle C\subseteq X}

292-406: A topology , which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets , which is easier than the others to manipulate. A topological space is the most general type of a mathematical space that allows for

365-964: A (possibly empty) set. The elements of X {\displaystyle X} are usually called points , though they can be any mathematical object. Let N {\displaystyle {\mathcal {N}}} be a function assigning to each x {\displaystyle x} (point) in X {\displaystyle X} a non-empty collection N ( x ) {\displaystyle {\mathcal {N}}(x)} of subsets of X . {\displaystyle X.} The elements of N ( x ) {\displaystyle {\mathcal {N}}(x)} will be called neighbourhoods of x {\displaystyle x} with respect to N {\displaystyle {\mathcal {N}}} (or, simply, neighbourhoods of x {\displaystyle x} ). The function N {\displaystyle {\mathcal {N}}}

438-442: A basic open set, all but finitely many of its projections are the entire space. A quotient space is defined as follows: if X {\displaystyle X} is a topological space and Y {\displaystyle Y} is a set, and if f : X → Y {\displaystyle f:X\to Y} is a surjective function , then the quotient topology on Y {\displaystyle Y}

511-399: A basis set consisting of all subsets of the union of the U i {\displaystyle U_{i}} that have non-empty intersections with each U i . {\displaystyle U_{i}.} The Fell topology on the set of all non-empty closed subsets of a locally compact Polish space X {\displaystyle X} is a variant of

584-427: A collection τ {\displaystyle \tau } of closed subsets of X . {\displaystyle X.} Thus the sets in the topology τ {\displaystyle \tau } are the closed sets, and their complements in X {\displaystyle X} are the open sets. There are many other equivalent ways to define a topological space: in other words

657-564: A homeomorphism between them. From the standpoint of topology, homeomorphic spaces are essentially identical. In category theory , one of the fundamental categories is Top , which denotes the category of topological spaces whose objects are topological spaces and whose morphisms are continuous functions. The attempt to classify the objects of this category ( up to homeomorphism ) by invariants has motivated areas of research, such as homotopy theory , homology theory , and K-theory . A given set may have many different topologies. If

730-454: A local point of view (as parametric surfaces) and topological issues were never considered". " Möbius and Jordan seem to be the first to realize that the main problem about the topology of (compact) surfaces is to find invariants (preferably numerical) to decide the equivalence of surfaces, that is, to decide whether two surfaces are homeomorphic or not." The subject is clearly defined by Felix Klein in his " Erlangen Program " (1872):

803-567: A particular sequence of functions converges to the zero function. A linear graph has a natural topology that generalizes many of the geometric aspects of graphs with vertices and edges . Outer space of a free group F n {\displaystyle F_{n}} consists of the so-called "marked metric graph structures" of volume 1 on F n . {\displaystyle F_{n}.} Topological spaces can be broadly classified, up to homeomorphism, by their topological properties . A topological property

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876-472: A real number x {\displaystyle x} if it includes an open interval containing x . {\displaystyle x.} Given such a structure, a subset U {\displaystyle U} of X {\displaystyle X} is defined to be open if U {\displaystyle U} is a neighbourhood of all points in U . {\displaystyle U.} The open sets then satisfy

949-403: A set B , also termed the set difference of B and A , written B ∖ A , {\displaystyle B\setminus A,} is the set of elements in B that are not in A . If A is a set, then the absolute complement of A (or simply the complement of A ) is the set of elements not in A (within a larger set that is implicitly defined). In other words, let U be

1022-473: A set is given a different topology, it is viewed as a different topological space. Any set can be given the discrete topology in which every subset is open. The only convergent sequences or nets in this topology are those that are eventually constant. Also, any set can be given the trivial topology (also called the indiscrete topology), in which only the empty set and the whole space are open. Every sequence and net in this topology converges to every point of

1095-862: A set may have many distinct topologies defined on it. If γ {\displaystyle \gamma } is an ordinal number , then the set γ = [ 0 , γ ) {\displaystyle \gamma =[0,\gamma )} may be endowed with the order topology generated by the intervals ( α , β ) , {\displaystyle (\alpha ,\beta ),} [ 0 , β ) , {\displaystyle [0,\beta ),} and ( α , γ ) {\displaystyle (\alpha ,\gamma )} where α {\displaystyle \alpha } and β {\displaystyle \beta } are elements of γ . {\displaystyle \gamma .} Every manifold has

1168-549: A set that contains all the elements under study; if there is no need to mention U , either because it has been previously specified, or it is obvious and unique, then the absolute complement of A is the relative complement of A in U : A ∁ = U ∖ A = { x ∈ U : x ∉ A } . {\displaystyle A^{\complement }=U\setminus A=\{x\in U:x\notin A\}.} The absolute complement of A

1241-456: A topology native to it, and this can be extended to vector spaces over that field. The Zariski topology is defined algebraically on the spectrum of a ring or an algebraic variety . On R n {\displaystyle \mathbb {R} ^{n}} or C n , {\displaystyle \mathbb {C} ^{n},} the closed sets of the Zariski topology are

1314-467: A topology. In the usual topology on R n {\displaystyle \mathbb {R} ^{n}} the basic open sets are the open balls . Similarly, C , {\displaystyle \mathbb {C} ,} the set of complex numbers , and C n {\displaystyle \mathbb {C} ^{n}} have a standard topology in which the basic open sets are open balls. For any algebraic objects we can introduce

1387-867: A universe U . The following identities capture notable properties of relative complements: A binary relation R {\displaystyle R} is defined as a subset of a product of sets X × Y . {\displaystyle X\times Y.} The complementary relation R ¯ {\displaystyle {\bar {R}}} is the set complement of R {\displaystyle R} in X × Y . {\displaystyle X\times Y.} The complement of relation R {\displaystyle R} can be written R ¯   =   ( X × Y ) ∖ R . {\displaystyle {\bar {R}}\ =\ (X\times Y)\setminus R.} Here, R {\displaystyle R}

1460-456: A universe U . The following identities capture important properties of absolute complements: De Morgan's laws : Complement laws: Involution or double complement law: Relationships between relative and absolute complements: Relationship with a set difference: The first two complement laws above show that if A is a non-empty, proper subset of U , then { A , A } is a partition of U . If A and B are sets, then

1533-641: Is finer than τ 1 , {\displaystyle \tau _{1},} and τ 1 {\displaystyle \tau _{1}} is coarser than τ 2 . {\displaystyle \tau _{2}.} A proof that relies only on the existence of certain open sets will also hold for any finer topology, and similarly a proof that relies only on certain sets not being open applies to any coarser topology. The terms larger and smaller are sometimes used in place of finer and coarser, respectively. The terms stronger and weaker are also used in

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1606-481: Is empty ). The interior and exterior of a closed curve are a slightly different concept; see the Jordan curve theorem . If S {\displaystyle S} is a subset of a Euclidean space , then x {\displaystyle x} is an interior point of S {\displaystyle S} if there exists an open ball centered at x {\displaystyle x} which

1679-488: Is a Baire space . The exterior of a subset S {\displaystyle S} of a topological space X , {\displaystyle X,} denoted by ext X ⁡ S {\displaystyle \operatorname {ext} _{X}S} or simply ext ⁡ S , {\displaystyle \operatorname {ext} S,} is the largest open set disjoint from S , {\displaystyle S,} namely, it

1752-589: Is a collection of topologies on X , {\displaystyle X,} then the meet of F {\displaystyle F} is the intersection of F , {\displaystyle F,} and the join of F {\displaystyle F} is the meet of the collection of all topologies on X {\displaystyle X} that contain every member of F . {\displaystyle F.} A function f : X → Y {\displaystyle f:X\to Y} between topological spaces

1825-520: Is a subset of a topological space X {\displaystyle X} then x {\displaystyle x} is an interior point of S {\displaystyle S} in X {\displaystyle X} if x {\displaystyle x} is contained in an open subset of X {\displaystyle X} that is completely contained in S . {\displaystyle S.} (Equivalently, x {\displaystyle x}

1898-561: Is ambiguous, as in some contexts (for example, Minkowski set operations in functional analysis ) it can be interpreted as the set of all elements b − a , {\displaystyle b-a,} where b is taken from B and a from A . Formally: B ∖ A = { x ∈ B : x ∉ A } . {\displaystyle B\setminus A=\{x\in B:x\notin A\}.} Let A , B , and C be three sets in

1971-658: Is an interior point of S {\displaystyle S} if S {\displaystyle S} is a neighbourhood of x . {\displaystyle x.} ) The interior of a subset S {\displaystyle S} of a topological space X , {\displaystyle X,} denoted by int X ⁡ S {\displaystyle \operatorname {int} _{X}S} or int ⁡ S {\displaystyle \operatorname {int} S} or S ∘ , {\displaystyle S^{\circ },} can be defined in any of

2044-539: Is an interior point of S {\displaystyle S} if there exists a real number r > 0 , {\displaystyle r>0,} such that y {\displaystyle y} is in S {\displaystyle S} whenever the distance d ( x , y ) < r . {\displaystyle d(x,y)<r.} This definition generalizes to topological spaces by replacing "open ball" with " open set ". If S {\displaystyle S}

2117-542: Is called continuous if for every x ∈ X {\displaystyle x\in X} and every neighbourhood N {\displaystyle N} of f ( x ) {\displaystyle f(x)} there is a neighbourhood M {\displaystyle M} of x {\displaystyle x} such that f ( M ) ⊆ N . {\displaystyle f(M)\subseteq N.} This relates easily to

2190-399: Is called a neighbourhood topology if the axioms below are satisfied; and then X {\displaystyle X} with N {\displaystyle {\mathcal {N}}} is called a topological space . The first three axioms for neighbourhoods have a clear meaning. The fourth axiom has a very important use in the structure of the theory, that of linking together

2263-398: Is completely contained in S . {\displaystyle S.} (This is illustrated in the introductory section to this article.) This definition generalizes to any subset S {\displaystyle S} of a metric space X {\displaystyle X} with metric d {\displaystyle d} : x {\displaystyle x}

Interior (topology) - Misplaced Pages Continue

2336-677: Is defined on the topological space X . {\displaystyle X.} The map f {\displaystyle f} is then the natural projection onto the set of equivalence classes . The Vietoris topology on the set of all non-empty subsets of a topological space X , {\displaystyle X,} named for Leopold Vietoris , is generated by the following basis: for every n {\displaystyle n} -tuple U 1 , … , U n {\displaystyle U_{1},\ldots ,U_{n}} of open sets in X , {\displaystyle X,} we construct

2409-425: Is empty): X = int ⁡ S ∪ ∂ S ∪ ext ⁡ S , {\displaystyle X=\operatorname {int} S\cup \partial S\cup \operatorname {ext} S,} where ∂ S {\displaystyle \partial S} denotes the boundary of S . {\displaystyle S.} The interior and exterior are always open , while

2482-504: Is generated by the open intervals . The set of all open intervals forms a base or basis for the topology, meaning that every open set is a union of some collection of sets from the base. In particular, this means that a set is open if there exists an open interval of non zero radius about every point in the set. More generally, the Euclidean spaces R n {\displaystyle \mathbb {R} ^{n}} can be given

2555-420: Is often viewed as a logical matrix with rows representing the elements of X , {\displaystyle X,} and columns elements of Y . {\displaystyle Y.} The truth of a R b {\displaystyle aRb} corresponds to 1 in row a , {\displaystyle a,} column b . {\displaystyle b.} Producing

2628-498: Is produced by \complement . (It corresponds to the Unicode symbol U+2201 ∁ COMPLEMENT .) Topological space In mathematics , a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance . More specifically, a topological space is a set whose elements are called points , along with an additional structure called

2701-460: Is said to be closed in ( X , τ ) {\displaystyle (X,\tau )} if its complement X ∖ C {\displaystyle X\setminus C} is an open set. Using de Morgan's laws , the above axioms defining open sets become axioms defining closed sets : Using these axioms, another way to define a topological space is as a set X {\displaystyle X} together with

2774-537: Is the topological space containing S , {\displaystyle S,} and the backslash ∖ {\displaystyle \,\setminus \,} denotes set-theoretic difference . Therefore, the abstract theory of closure operators and the Kuratowski closure axioms can be readily translated into the language of interior operators, by replacing sets with their complements in X . {\displaystyle X.} In general,

2847-424: Is the collection of subsets of Y {\displaystyle Y} that have open inverse images under f . {\displaystyle f.} In other words, the quotient topology is the finest topology on Y {\displaystyle Y} for which f {\displaystyle f} is continuous. A common example of a quotient topology is when an equivalence relation

2920-540: Is the union of all open sets in X {\displaystyle X} that are disjoint from S . {\displaystyle S.} The exterior is the interior of the complement, which is the same as the complement of the closure; in formulas, ext ⁡ S = int ⁡ ( X ∖ S ) = X ∖ S ¯ . {\displaystyle \operatorname {ext} S=\operatorname {int} (X\setminus S)=X\setminus {\overline {S}}.} Similarly,

2993-447: Is usually denoted by A ∁ {\displaystyle A^{\complement }} . Other notations include A ¯ , A ′ , {\displaystyle {\overline {A}},A',} ∁ U A ,  and  ∁ A . {\displaystyle \complement _{U}A,{\text{ and }}\complement A.} Let A and B be two sets in

Interior (topology) - Misplaced Pages Continue

3066-606: Is usually used for rendering a set difference symbol, which is similar to a backslash symbol. When rendered, the \setminus command looks identical to \backslash , except that it has a little more space in front and behind the slash, akin to the LaTeX sequence \mathbin{\backslash} . A variant \smallsetminus is available in the amssymb package, but this symbol is not included separately in Unicode. The symbol ∁ {\displaystyle \complement } (as opposed to C {\displaystyle C} )

3139-431: The cofinite topology in which the open sets are the empty set and the sets whose complement is finite. This is the smallest T 1 topology on any infinite set. Any set can be given the cocountable topology , in which a set is defined as open if it is either empty or its complement is countable. When the set is uncountable, this topology serves as a counterexample in many situations. The real line can also be given

3212-447: The complement of a set A , often denoted by A ∁ {\displaystyle A^{\complement }} (or A ′ ), is the set of elements not in A . When all elements in the universe , i.e. all elements under consideration, are considered to be members of a given set U , the absolute complement of A is the set of elements in U that are not in A . The relative complement of A with respect to

3285-483: The formula V − E + F = 2 {\displaystyle V-E+F=2} relating the number of vertices (V), edges (E) and faces (F) of a convex polyhedron , and hence of a planar graph . The study and generalization of this formula, specifically by Cauchy (1789–1857) and L'Huilier (1750–1840), boosted the study of topology. In 1827, Carl Friedrich Gauss published General investigations of curved surfaces , which in section 3 defines

3358-495: The lower limit topology . Here, the basic open sets are the half open intervals [ a , b ) . {\displaystyle [a,b).} This topology on R {\displaystyle \mathbb {R} } is strictly finer than the Euclidean topology defined above; a sequence converges to a point in this topology if and only if it converges from above in the Euclidean topology. This example shows that

3431-480: The relative complement of A in B , also termed the set difference of B and A , is the set of elements in B but not in A . The relative complement of A in B is denoted B ∖ A {\displaystyle B\setminus A} according to the ISO 31-11 standard . It is sometimes written B − A , {\displaystyle B-A,} but this notation

3504-505: The solution sets of systems of polynomial equations. If Γ {\displaystyle \Gamma } is a filter on a set X {\displaystyle X} then { ∅ } ∪ Γ {\displaystyle \{\varnothing \}\cup \Gamma } is a topology on X . {\displaystyle X.} Many sets of linear operators in functional analysis are endowed with topologies that are defined by specifying when

3577-466: The 1930s, James Waddell Alexander II and Hassler Whitney first expressed the idea that a surface is a topological space that is locally like a Euclidean plane . Topological spaces were first defined by Felix Hausdorff in 1914 in his seminal "Principles of Set Theory". Metric spaces had been defined earlier in 1906 by Maurice Fréchet , though it was Hausdorff who popularised the term "metric space" ( German : metrischer Raum ). The utility of

3650-475: The Vietoris topology, and is named after mathematician James Fell. It is generated by the following basis: for every n {\displaystyle n} -tuple U 1 , … , U n {\displaystyle U_{1},\ldots ,U_{n}} of open sets in X {\displaystyle X} and for every compact set K , {\displaystyle K,}

3723-544: The axioms given below in the next definition of a topological space. Conversely, when given the open sets of a topological space, the neighbourhoods satisfying the above axioms can be recovered by defining N {\displaystyle N} to be a neighbourhood of x {\displaystyle x} if N {\displaystyle N} includes an open set U {\displaystyle U} such that x ∈ U . {\displaystyle x\in U.} A topology on

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3796-419: The basic open sets are open balls defined by the metric. This is the standard topology on any normed vector space . On a finite-dimensional vector space this topology is the same for all norms. There are many ways of defining a topology on R , {\displaystyle \mathbb {R} ,} the set of real numbers . The standard topology on R {\displaystyle \mathbb {R} }

3869-425: The boundary is closed . Some of the properties of the exterior operator are unlike those of the interior operator: Two shapes a {\displaystyle a} and b {\displaystyle b} are called interior-disjoint if the intersection of their interiors is empty. Interior-disjoint shapes may or may not intersect in their boundary. Absolute complement In set theory ,

3942-472: The complementary relation to R {\displaystyle R} then corresponds to switching all 1s to 0s, and 0s to 1s for the logical matrix of the complement. Together with composition of relations and converse relations , complementary relations and the algebra of sets are the elementary operations of the calculus of relations . In the LaTeX typesetting language, the command \setminus

4015-550: The concept of sequence . A topology is completely determined if for every net in X {\displaystyle X} the set of its accumulation points is specified. Many topologies can be defined on a set to form a topological space. When every open set of a topology τ 1 {\displaystyle \tau _{1}} is also open for a topology τ 2 , {\displaystyle \tau _{2},} one says that τ 2 {\displaystyle \tau _{2}}

4088-454: The concept of a topology is shown by the fact that there are several equivalent definitions of this mathematical structure . Thus one chooses the axiomatization suited for the application. The most commonly used is that in terms of open sets , but perhaps more intuitive is that in terms of neighbourhoods and so this is given first. This axiomatization is due to Felix Hausdorff . Let X {\displaystyle X} be

4161-405: The concepts of neighbourhood, or that of open or closed sets can be reconstructed from other starting points and satisfy the correct axioms. Another way to define a topological space is by using the Kuratowski closure axioms , which define the closed sets as the fixed points of an operator on the power set of X . {\displaystyle X.} A net is a generalisation of

4234-447: The curved surface in a similar manner to the modern topological understanding: "A curved surface is said to possess continuous curvature at one of its points A, if the direction of all the straight lines drawn from A to points of the surface at an infinitesimal distance from A are deflected infinitesimally from one and the same plane passing through A." Yet, "until Riemann 's work in the early 1850s, surfaces were always dealt with from

4307-441: The definition of limits , continuity , and connectedness . Common types of topological spaces include Euclidean spaces , metric spaces and manifolds . Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological spaces in their own right is called point-set topology or general topology . Around 1735, Leonhard Euler discovered

4380-416: The discrete topology, under which the algebraic operations are continuous functions. For any such structure that is not finite, we often have a natural topology compatible with the algebraic operations, in the sense that the algebraic operations are still continuous. This leads to concepts such as topological groups , topological vector spaces , topological rings and local fields . Any local field has

4453-436: The following equivalent ways: If the space X {\displaystyle X} is understood from context then the shorter notation int ⁡ S {\displaystyle \operatorname {int} S} is usually preferred to int X ⁡ S . {\displaystyle \operatorname {int} _{X}S.} On the set of real numbers , one can put other topologies rather than

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4526-417: The geometry invariants of arbitrary continuous transformation, a kind of geometry. The term "topology" was introduced by Johann Benedict Listing in 1847, although he had used the term in correspondence some years earlier instead of previously used "Analysis situs". The foundation of this science, for a space of any dimension, was created by Henri Poincaré . His first article on this topic appeared in 1894. In

4599-418: The interior is the exterior of the complement: int ⁡ S = ext ⁡ ( X ∖ S ) . {\displaystyle \operatorname {int} S=\operatorname {ext} (X\setminus S).} The interior, boundary , and exterior of a set S {\displaystyle S} together partition the whole space into three blocks (or fewer when one or more of these

4672-463: The interior operator does not commute with unions. However, in a complete metric space the following result does hold: Theorem   (C. Ursescu)  —  Let S 1 , S 2 , … {\displaystyle S_{1},S_{2},\ldots } be a sequence of subsets of a complete metric space X . {\displaystyle X.} The result above implies that every complete metric space

4745-489: The literature, but with little agreement on the meaning, so one should always be sure of an author's convention when reading. The collection of all topologies on a given fixed set X {\displaystyle X} forms a complete lattice : if F = { τ α : α ∈ A } {\displaystyle F=\left\{\tau _{\alpha }:\alpha \in A\right\}}

4818-402: The neighbourhoods of different points of X . {\displaystyle X.} A standard example of such a system of neighbourhoods is for the real line R , {\displaystyle \mathbb {R} ,} where a subset N {\displaystyle N} of R {\displaystyle \mathbb {R} } is defined to be a neighbourhood of

4891-422: The open sets of the larger space with the subset. For any indexed family of topological spaces, the product can be given the product topology , which is generated by the inverse images of open sets of the factors under the projection mappings. For example, in finite products, a basis for the product topology consists of all products of open sets. For infinite products, there is the additional requirement that in

4964-515: The sense that int X ⁡ S = X ∖ ( X ∖ S ) ¯ {\displaystyle \operatorname {int} _{X}S=X\setminus {\overline {(X\setminus S)}}} and also S ¯ = X ∖ int X ⁡ ( X ∖ S ) , {\displaystyle {\overline {S}}=X\setminus \operatorname {int} _{X}(X\setminus S),} where X {\displaystyle X}

5037-415: The set of all subsets of X {\displaystyle X} that are disjoint from K {\displaystyle K} and have nonempty intersections with each U i {\displaystyle U_{i}} is a member of the basis. Metric spaces embody a metric , a precise notion of distance between points. Every metric space can be given a metric topology, in which

5110-458: The space. This example shows that in general topological spaces, limits of sequences need not be unique. However, often topological spaces must be Hausdorff spaces where limit points are unique. There exist numerous topologies on any given finite set . Such spaces are called finite topological spaces . Finite spaces are sometimes used to provide examples or counterexamples to conjectures about topological spaces in general. Any set can be given

5183-538: The standard one: These examples show that the interior of a set depends upon the topology of the underlying space. The last two examples are special cases of the following. Let X {\displaystyle X} be a topological space and let S {\displaystyle S} and T {\displaystyle T} be subsets of X . {\displaystyle X.} Other properties include: Relationship with closure The above statements will remain true if all instances of

5256-468: The symbols/words are respectively replaced by and the following symbols are swapped: For more details on this matter, see interior operator below or the article Kuratowski closure axioms . The interior operator int X {\displaystyle \operatorname {int} _{X}} is dual to the closure operator, which is denoted by cl X {\displaystyle \operatorname {cl} _{X}} or by an overline , in

5329-414: The usual definition in analysis. Equivalently, f {\displaystyle f} is continuous if the inverse image of every open set is open. This is an attempt to capture the intuition that there are no "jumps" or "separations" in the function. A homeomorphism is a bijection that is continuous and whose inverse is also continuous. Two spaces are called homeomorphic if there exists

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